The problem I am working on is:
Use a Venn diagram to illustrate the relationship $A⊆B$ and $B⊆C$.
From my understanding, I should be drawing several Venn diagrams, corresponding to the different situations that are possible.
Since $A \subseteq B$ does not specify whether $A$ is a proper subset or not, we have two situations:
$(1)$: $A=B$
$(2)$: $A \subset B$
The same two situations apply to $B \subseteq C$
(This part is just a side note, but I'd appreciate a corroboration of my reasoning)
Also, by the transitive law, since for x we find in A, it is true that we can find it in B ($A \subseteq B$); and since for every x in B, it is true that we can find it in C ($B \subseteq C$), we can say that $A \subseteq C$. This is reasonable, because all of A is contained in B, and all of B is contained in C, so all of A must be contained in C.
Now, the different situations we have are as follows:
$(1)$: $A=B$ and $B \subset C$
$(2)$: $A \subset C$ and $B \subset C$
$(3)$: $A=B$ and $B=C$
$(4)$: $A \subset C$ and $B=C$
So, I would have to make a Venn diagram for each situation?
Best Answer
Here's the list I came up with, with the "$\subsetneq$" symbol meaning "is a proper subset of":
$(1): A = B\;$ and $\;B = C.\;$ (And hence $A = C$ by transitivity)
$(2): A = B\;$ and $\;B \subsetneq C.\;$ (And hence $A \subsetneq C$, necessarily).
$(3): A \subsetneq B\;$ and $\; B = C.\;$ (And hence, $A \subsetneq C$, necessarily).
$(4): A \subsetneq B\;$ and $\; B \subsetneq C.\;$ (And hence, $A \subsetneq C$, necessarily).
Yes, the Venn diagram for each of the four scenarios would necessarily be different, if we are distinguishing between equality of sets and "being a proper subset of" a set, so you can cover all possible relationships by drawing a Venn diagram for each case, separately. For example, case $(1)$ would be a single circle, labeled A = B = C. Case two would be a circle A = B, contained within the larger circle C. etc...